Finite Group Databasebeta

← #46846274

Subgroups — #46846274

Order 34688.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774407C211yesmaximal, characteristic
2C2×#22576118C211yesmaximal, characteristic, complemented
3#31774241C212yesmaximal, complemented
4#31774232C211yesmaximal, characteristic, complemented
5#31774228C211yesmaximal, characteristic, complemented
6#31774220C212yesmaximal, complemented
7#31774208C211yesmaximal, characteristic, complemented
8#31773932C212yesmaximal, complemented
9#31773890C212yesmaximal, complemented
10C22⋊C8×D271C211yesmaximal, characteristic, complemented
11#481×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2211yescharacteristic
13C22×C4×D271C2211yescharacteristic
14C2×C8×D271C2212yes—
15#22576142—21no—
16#22576140C2211yescharacteristic
17#22576139C2211yescharacteristic
18#22576131C2211yescharacteristic
19#22576128—21no—
20D4×Dic271—21no—
21(C22⋊D271)⋊C4—21no—
22#22576118—21no—
23#22576118—21no—
24#22576118C2212yes—
25#22576118C2212yes—
26#22576118C2211yescharacteristic
27C2×C4⋊Dic271C2211yescharacteristic
28C2×(C2×D271)⋊C4—21no—
29C2×C4×Dic271—21no—
30#22576093C2212yescomplemented
31#22576093C2212yescomplemented
32#22576092C2212yescomplemented
33#22576085C2212yescomplemented
34#22576083C2212yescomplemented
35#22576081C2212yescomplemented
36C22⋊(C271⋊C8)C2211yescharacteristic
37(C2×D271)⋊C8C2212yes—
38#121×C271C2211yescharacteristic
39C2×C4⋊C4×C271C2211yescharacteristic
40#106×C271C2212yescomplemented
41#102×C271C2212yescomplemented
42C22⋊C8×C271C2211yescharacteristic
43C23×D271D411yescharacteristic
44#16864653—24no—
45C2×Dic542C2311yescharacteristic
46C2×C22⋊D271—21no—
47C2×C22⋊D271—21no—
48C22×Dic271—21no—
49C22×Dic271—21no—
50C22×Dic271D411yescharacteristic
51C2×C4×D271—24no—
52C2×C4×D271D411yescharacteristic
53C2×C4×D271D411yescharacteristic
54C2×C4×D271C2311yescharacteristic
55C2×C4×D271C2311yescharacteristic
56C2×C271⋊C8C2312yes—
57C8×D271—24no—
58C4⋊Dic271—21no—
59C4⋊Dic271—21no—
60C4⋊Dic271C2311yescharacteristic
61C4⋊Dic271C2312yes—
62Dic271⋊C4—21no—
63C22⋊Dic271—21no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—21no—
66(C2×D271)⋊C4—21no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—22no—
69C4×Dic271—22no—
70C4×Dic271—22no—
71C4×Dic271—21no—
72C2×D4×C271—21no—
73C2×D4×C271C2311yescharacteristic
74C22×C4×C271—21no—
75C22×C4×C271C2311yescharacteristic
76C2×C8×C271C2312yes—
77C4⋊C4×C271—21no—
78C4⋊C4×C271C2311yescharacteristic
79C4⋊C4×C271C2312yes—
80C22⋊C4×C271—21no—
81C22×D271—22no—
82C22×D271—22no—
83C22×D271—41no—
84C22×D271C2×D411yescharacteristic
85C22×D271C2×D411yescharacteristic
86Dic542—22no—
87C22⋊D271—42no—
88C22⋊D271—42no—
89C2×Dic271—22no—
90C2×Dic271—22no—
91C2×Dic271—21no—
92C2×Dic271—21no—
93C2×Dic271—42no—
94C2×Dic271—42no—
95C2×Dic271C2×D411yescharacteristic
96C2×Dic271C2×D411yescharacteristic
97C4×D271—22no—
98C4×D271—22no—
99C4×D271—22no—
100C4×D271#4114yes—
101C271⋊C8—22no—
102C23×C271—21no—
103C23×C271C2×D411yescharacteristic
104D4×C271—22no—
105D4×C271—42no—
106C2×C4×C271—22no—
107C2×C4×C271—21no—
108C2×C4×C271—22no—
109C2×C4×C271—42no—
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—22no—
113C22×C271—21no—
114C22×C271—21no—
115C22×C271—41no—
116C22×C271—41no—
117C22×C271—41no—
118C22×C271#13512yes—
119C22×C271C22×D411yescharacteristic
120C2×D271—21no—
121C2×D271—21no—
122C2×D271—21no—
123C2×D271—21no—
124C2×D271—21no—
125C2×D271—44no—
126C2×D271#12312yes—
127C4×C271—21no—
128C4×C271—41no—
129C4×C271—42no—
130C4×C271C2×#4112yes—
131Dic271—21no—
132Dic271—41no—
133Dic271—42no—
134Dic271—41no—
135Dic271#12312yes—
136C2×C271—22no—
137C2×C271—81no—
138C2×C271C2×#12311yescharacteristic
139C2×C271C2×#13511yescharacteristic
140C2×C271#57511yescharacteristic
141D271—22no—
142D271—41no—
143C271#309111yescharacteristic, complemented, Sylow
144#3091—2711nomaximal, Sylow
145#525—2711no—
146C2×#117—2711no—
147#485—2712no—
148#484—2711no—
149#483—2711no—
150#482—2712no—
151#481D27111yescharacteristic, complemented
152#480—2711no—
153#427—2712no—
154#416—2712no—
155C2×C22⋊C8—2711no—
156C2×#41—2711no—
157C23×C4—2711no—
158C22×C8—2712no—
159#123—5421no—
160C22⋊Q8—2711no—
161C22⋊Q8—2711no—
162#121—2711no—
163#121C2×D27111yescharacteristic
164#120—5421no—
165C4×D4—5421no—
166C4×D4—5421no—
167#117—2712no—
168#117—2712no—
169#117—2711no—
170#117—5421no—
171#117—5421no—
172C2×C4⋊C4—2711no—
173C2×C4⋊C4C2×D27111yescharacteristic
174C2×C22⋊C4—5421no—
175C2×C42—5421no—
176#107—2712no—
177#107—2712no—
178#106—2712no—
179#106C2×D27112yescomplemented
180#103—2712no—
181#103—2712no—
182#102—2712no—
183#102C2×D27112yescomplemented
184C22⋊C8—2711no—
185C22⋊C8—2712no—
186C22⋊C8C2×D27111yescharacteristic
187C24—2711no—
188#41—5424no—
189C2×Q8—2711no—
190C2×D4—21no—
191C2×D4—5421no—
192C2×D4—5421no—
193C2×D4C22×D27111yescharacteristic
194C22×C4—21no—
195C22×C4—2711no—
196C22×C4—2711no—
197C22×C4—2711no—
198C22×C4—2711no—
199C22×C4—2711no—
200C22×C4—5424no—
201C22×C4—5421no—
202C22×C4—5421no—
203C22×C4C22×D27111yescharacteristic
204C2×C8—2712no—
205C2×C8—5424no—
206C2×C8C22×D27112yes—
207C4⋊C4—21no—
208C4⋊C4—2712no—
209C4⋊C4—2711no—
210C4⋊C4—5421no—
211C4⋊C4—5421no—
212C4⋊C4—5421no—
213C4⋊C4C22×D27112yes—
214C4⋊C4C22×D27111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—5422no—
217C22⋊C4—5422no—
218C22⋊C4—5421no—
219C22⋊C4—5421no—
220C22⋊C4—5421no—
221C22⋊C4—5421no—
222C42—5422no—
223C42—5422no—
224C42—5421no—
225C23—21no—
226C23—2711no—
227C23—2711no—
228C23—5422no—
229C23—5422no—
230C23—10841no—
231C23D4×D27111yescharacteristic
232Q8—5422no—
233D4—22no—
234D4—42no—
235D4—10842no—
236D4—10842no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—2711no—
242C2×C4—2714no—
243C2×C4—2711no—
244C2×C4—5422no—
245C2×C4—5422no—
246C2×C4—5422no—
247C2×C4—5422no—
248C2×C4—5422no—
249C2×C4—5421no—
250C2×C4—5421no—
251C2×C4—10842no—
252C2×C4—10842no—
253C2×C4D4×D27111yescharacteristic
254C2×C4C23×D27111yescharacteristic
255C8—22no—
256C8—5422no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—2712no—
263C22—5421no—
264C22—5421no—
265C22—5421no—
266C22—5421no—
267C22—5421no—
268C22—10844no—
269C22#2257618112yes—
270C22C2×D4×D27111yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—2712no—
275C4—5421no—
276C4—10841no—
277C4—10842no—
278C4—10841no—
279C4#41×D27112yes—
280C2—22no—
281C2—81no—
282C2—5422no—
283C2—10841no—
284C2#123×D27111yescharacteristic
285C2C2×#2257618111yescharacteristic
286C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#481×C271C2
socleC22×C271C22×D4
prime radical#31774407C2
hypercenter#481D271
hypocenterC271#3091
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#22576118C2semidirect
3. #31774241C2semidirect
4. #31774232C2semidirect
5. #31774228C2semidirect
6. #31774220C2semidirect
7. #31774208C2semidirect
8. #31773932C2semidirect
9. #31773890C2semidirect
10. C22⋊C8×D271C2semidirect
11. #481×C271C2semidirect
30. #22576093C22semidirect
31. #22576093C22semidirect
32. #22576092C22semidirect
33. #22576085C22semidirect
34. #22576083C22semidirect
35. #22576081C22semidirect
40. #106×C271C22semidirect
41. #102×C271C22semidirect
143. C271#3091semidirect
151. #481D271semidirect
179. #106C2×D271semidirect
183. #102C2×D271semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317744072
C2×#225761182
#317742412
#317742322
#317742282
#317742202
#317742082
#317739322
#317738902
C22⋊C8×D2712
#481×C2712
#3091271